Union of Sets¶

Definition¶

Given two sets $A$ and $B\text{,}$ the union $A\cup B$ of $A$ and $B$ is a set whose elements consist of all the elements which belong to $A$ or $B$ or both,

x\in A\cup B \; \iff \; x\in A \text{ or } x\in B \iff \; (x\in A)\lor (x\in B).

Interpretation¶

The union of sets combines elements from multiple collections. The "or" in the definition is inclusive: an element that lies in both sets belongs to the union, and it appears there only once. Union is the set version of the logical "or", just as intersection is the set version of "and".

Two overlapping discs A and B with their whole combined area shaded as the union.

The whole area covered by either disc is shaded. The lens in the middle lies in both $A$ and $B$ , and it belongs to the union once, not twice. Each disc lies inside the shaded region, so $A\subseteq A\cup B$ and $B\subseteq A\cup B$ $B$ . If $A\subseteq B$ $B$ , the union adds nothing new: $A\cup B=B$ if and only if $A\subseteq B$ $B$ .

A common slip is to count elements as $|A\cup B|=|A|+|B|$ $∣ B ∣$ . This holds only for disjoint finite sets; in general the overlap is counted twice and must be subtracted once, $|A\cup B|=|A|+|B|-|A\cap B|$ $B ∣$ . For $\{1,2\}\cup\{2,3\}$ this gives $2+2-1=3$ $3$ .

Key Results¶

Elements belong to the union if they belong to at least one set.
Union is commutative and associative, i.e., we have $A\cup B=B\cup A$ and $A\cup(B\cup C)=(A\cup B)\cup C\text{.}$
The union with the empty set leaves a set unchanged.

Examples¶

$\{1,\,2\}\cup\{2,\,3\}=\{1,\,2,\,3\}\text{.}$
$A\cup\varnothing=A\text{.}$

Prerequisites¶

Set

Sources¶

Math I: Calculus · Lecture 1

Aliases: set union Symbols: element of, intersection, subset, union

Last updated: 22 October 2026 19:04